Draw the graph for y = x^2 + 4x +2

This is a quadratic equation. So we know the shape will look something like this:
(draw 'smiley face' shape. Plot some points showing for x=2, y = 4. For x =3, y =9. Show symmetry: a negative number squared gives a positive).
Let's try factorising into 2 brackets. If we do this we can see where the graph cuts the x axis - we already know that it cuts the y axis at y = 2 by putting x = 0 in the equation.
To factorise, we want something in the format (x + a)(x + b) - remember, 'a' and/or 'b' can be negative too!The 'units' term in our equation is 2. Therefore a*b = 2. Our only options as factors of 2 are: 2 and 1. -2 and -1. These will not give us the 4x term. So this method doesn't work :(
That's ok! There is another method we can use. Let's try completing the square. This means putting the equation in the format y = (x + a)^2 + b. For 'a' we always take as HALF of our x coefficient.
(show expansion of (x+2)^2, show that we need to take away 4 then add our unit coefficient, 2).
So we have our equation in this form: y = (x+2)^2 - 2. Now we can find where it cuts the x axis by setting y to zero!
(show x+2 = sqrt(2) therefore cuts x axis at x = -2 +/-sqrt(2): highlight that a sqrt can be positive or negative).
So now we know where the graph cuts the x and y axis. Where is the minimum point? This is the lowest value of y our equation can have. Because our (x+2)^2 term is squared, it will always be positive, so we will get the minimum value of this part by substituting x = -2, making the term zero. This leaves us with our minimum point at (-2, -2).
That's our graph sketched! However... there is another quick way to look at it:In our completed square form we have y = (x+2)^2 - 2. This is equivalent to taking an x^2 graph and shifting it 2 spaces LEFT and 2 spaces DOWN. The first bit is confusing: if it says x PLUS 2 why should it move to the left, in a negative direction?
(draw x^2 graph. Take a point, (0,0) For the function (x+2)^2 we want the value of (0+2)^2 for our graph, which is 4. This is the same as the value for x =2 in the x^2 graph, so our points need to shift 2 points to the right for this translation).
As for the y intercept, it is more intuitive: a - 2 at the end of our 'completed square' equation indicates the graph should be shifted down by two points.
To conclude, if you can factorise a quadratic equation, it is slightly quicker to sketch the graph. However, completing the square is a quick and useful way to find the minimum point and rough location of the graph based on the translation of x^2.